When we define complex numbers we start creating a field on R2. I think that is impossible to do the same thing on R3 and, in general, on Rn. Do a theorem exist which prove that?
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1https://en.wikipedia.org/wiki/Frobenius_theorem_(real_division_algebras) – Arthur Mar 08 '19 at 17:00
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1No. See https://mathoverflow.net/questions/9014/field-structure-for-rn – John Douma Mar 08 '19 at 17:00
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1see https://math.stackexchange.com/questions/1784166/why-are-there-no-triernions-3-dimensional-analogue-of-complex-numbers-quate?noredirect=1&lq=1 – Robert Z Mar 08 '19 at 17:01